Exercise 3. Let Inf(N) be the set of infinite subsets of N. Prove that Inf(N) P(N). Exercise 4. Let D(N) be the set of weakly decreasing functions f : N → N; ie. those functions f such that f(n) ≥ f(m) for all 0 ≤ n < m. Prove that D(N) ≈ N.

If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?

Let R be the set of all infinite sequences of real numbers, with the operations u+v=(u1,u2,u3,......)+(v1,v2,v3,......)=(u1+v1,u2+v2,u3+v3,.....) and cu=c(u1,u2,u3,......)=(cu1,cu2,cu3,......). Determine whether R is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.

What does it mean to say that f(4) is a local maximum valueof f ?